This talk will be a gentle introduction to the relationship between rational points of an algebraic curve and the structure of its (algebraic) fundamental group. For hyperbolic curves over a finitely generated infinite field, Grothendieck conjectured (in 1983) a precise relationship between fundamental groups and rational points, which we will explain by analogy with the corresponding problem in topology. I will then discuss the degree to which Grothendieck's conjecture holds for the "generic curve of type (g,n)". For such curves, the structure of the mapping class group of type (g,n) plays a central role. Putman's recent result on Picard groups of moduli spaces of curves is one of the important ingredients needed to understand the fundamental group of generic curves and their relationship to rational points.

4:00 pm Wednesday, March 9, 2011Geometry-Analysis Seminar: Regularity of the entropy for random walks on free groups.
by Francois Ledrappier in HB 227

"We consider random walks on a free group. We let the directing probability vary among the ones with a fixed generating finite support. We prove that the entropy of the random walk and the linear drift are real analytic functions of the probability."

4:00 pm Thursday, March 10, 2011Colloquium: The index of an algebraic variety
by Dino Lorenzini (University of Georgia (Athens)) in HB 227

The index is a positive integer attached to a system of polynomial equations over a field K. If the system has a solution in K, the index is 1. We will recall the definition of the index, and survey some results in the field.

Thanks to works by M. Kontsevich and A. Zorich followed by C. Boissy, we have a classification of all Rauzy Classes of any given genus. It follows from these works that Rauzy Classes are closed under the operation of inverting the permutation. In this paper, we shall prove the existence of self-inverse permutations in every Rauzy Class by giving an explicit construction of such an element satisfying the sufficient conditions. As a corollary, we will give another proof that every Rauzy Class is closed under taking inverses. In the case of generalized permutations, generalized Rauzy Classes have been classified by works of M. Kontsevich, H. Masur and J. Smillie, E. Lanneau, and again C. Boissy. We state the definition of self-inverse for generalized permutations and prove a necessary and sufficient condition for a generalized Rauzy Class to contain self-inverse elements.

There is experimental evidence that in density-stratified fluids, contaminants can become trapped near the pycnocline, the interface below the fluid surface where the density changes sharply. Such experiments provide one explanation for the underwater "oil plumes" observed following the recent Deepwater Horizon oil leak in the Gulf of Mexico. Modeling the dynamics of deformations of the pycnocline (so-called internal waves) is a step toward providing a description of how the contaminants move vertically in the fluid column. Ideally, one would like to accurately and easily predict whether contaminants will interact with biological populations at various depths, and perhaps whether contaminants will reach the surface mixing layer. The Benjamin-Ono equation is a model for gravity-driven internal waves in certain density-stratified fluids. It has the features of being a nonlocal equation (the dispersion term involves the Hilbert transform of the disturbance profile) and also of having a Lax pair and an associated inverse-scattering algorithm for the solution of the Cauchy initial-value problem. We will review known phenomena associated with this equation in the limit when the dispersive effects are nominally small, and compare with the better-known Korteweg-de Vries (KdV) equation. Then we will present a new result establishing the zero-dispersion limit of the solution of the Benjamin-Ono Cauchy problem for certain initial data, in the topology of weak convergence. Our methodology is a novel analogue of the Lax-Levermore method in which the equilibrium measure is given more-or-less explicitly rather than via the solution of a variational problem. The proof relies on aspects of the method of moments from probability theory. This is joint work with Zhengjie Xu. The weak limit is given by a remarkably simple formula that is easy to implement, far easier than the analogous formula for the KdV equation. As it is a weak limit, it only captures the local mean value of wild oscillations that can form as a result of dispersive regularization of shock waves. It remains an open problem to rigorously obtain formulae for the upper and lower envelopes of the oscillatory wave packet, a result that would certainly have further application in the modeling of internal waves.

I will present recent joint work with Jan Metzger. A basic question in mathematical relativity is how geometric properties of an asymptotically flat manifold (or initial data set) encode information about the physical properties of the space time that it is embedded in. For example, the square root of the area of the outermost minimal surface of an initial data with non-negative scalar curvature provides a lower bound for the "mass" of its associated space time, as was conjectured by Penrose and proven by Bray and Huisken-Ilmanen. Other special surfaces that have been studied in this context include stable constant mean curvature surfaces or isoperimetric surfaces. I will explain why positive mass works to the effect that large stable constant mean curvature surfaces are always isoperimetric. This answers a question of Bray's and complements the results by Huisken-Yau and Qing-Tian on the "global uniqueness problem for stable CMC surfaces" in initial data sets with positive scalar curvature.

Abstract: Rigid cohomology is one flavor of Weil cohomology. This entails for instance that one can asociate to a scheme X over F_p a collection H^i(X) of finite dimensional Q_p-vector spaces (and variants with supports in a closed subscheme or compact support), which enjoy lots and lots of nice properties (e.g. functorality, excision, Gysin, duality, a trace formula -- basically everything one needs to give a proof of the Weil conjectures). Classically, the construction of rigid cohomology is a bit complicated and requires many choices, so that proving things like functorality (or even that it is well defined) are theorems in their own right. An important recent advance is the construction by le Stum of an `Overconvergent site' which computes the rigid cohomology of X. This site involves no choices and so it trivially well defined, and many things (like functorality) become transparent. In this talk I'll explain a bit about classical rigid cohomology and the overconvergent site (beginning with an exposition of characteristic 0 analogues), and explain some new work generalizing rigid cohomology to algebraic stacks (as well as why one would want to do such a thing).

The talk is devoted to the problem of ergodic decomposition for measures quasi-invariant under actions of inductively compact groups, for instance, the infinite symmetric or the infinite unitary group. As a simple case, let us first consider a continuous transformation of a compact metric space. The space of probability measures invariant under the transformation is itself a compact metric space, and its extremal points are exactly ergodic probability measures. Choquet's Theorem now allows one to represent a given invariant probability measure as an integral over the space of ergodic probability measures. Such a representation is called an ergodic decomposition. For actions of locally compact groups with a quasi-invariant measure, ergodic decompositions using Choquet's Theorem have been constructed by Greschonig and Schmidt. Note, however, that inductively compact groups are not locally compact. Rohlin proposed a different way of constructing ergodic decompositions using his theory of measurable partitions. The advantage of Rohlin's construction is that it does not make any topological assumptions and can be applied to any measurable automorphism of a Lebesgue probability space. On the other hand, Rohlin's construction, unlike a Choquet-type construction, essentially relies on the ergodic theorem. In the talk, Rohlin's approach will be used to establish existence and uniqueness of ergodic decompositions for measurable actions of inductively compact groups with a quasi-invariant measure. Instead of the ergodic theorem, the martingale convergence theorem is used.

Moment-angle manifolds and complexes are spaces acted on by a torus and parametrised by finite simplicial complexes. They are central objects in toric topology, and currently gaining much interest in the homotopy theory. Due the their combinatorial origins, moment-angle complexes also find applications in combinatorial geometry and commutative algebra. We describe several interpretations of moment-angle manifolds and complexes, including the intersections of quadrics, complements of subspace arrangements and level sets of moment maps. We overview the known results on the topology of moment-angle complexes, including the description of their cohomology rings, as well as the homotopy and diffeomorphism types in some particular cases. We also discuss complex-analytic structures on moment-angle manifolds and methods for calculating invariants of these structures.

Bounds on Weyl sums (exponential sums of polynomial sequences) are an important classical topic in analytic number theory with at least a century of history (Hardy and Littlewood, Weyl, Vinogradov, Hua, Vaughan, Wooley and many others). In our joint work with L. Flaminio, following Furstenberg we approach the problem as a question on the equidistribution of nilflows (or linear skew-shifts). Our results are not far behind the best results obtained only very recently by number theoretical methods (Wooley, 2011) and we hope that our methods may yield further progress. Our work is based on ideas from the theory of dynamical systems such as finding solutions of cohomological equations, invariant distributions and renormalization, and it is part of a more general program to develop a theory of weakly chaotic, parabolic systems. Interval exchange transformations or translation flows, horocycle flows and nilflows are the main examples and our work is in fact an attempt at generalizing the Kontsevich-Zorich picture for the deviation of ergodic averages of interval exchange transformations.

I will discuss the proof and consequences of the following result: if D and D' are reduced, alternating diagrams for a pair of links with branched double covers Y and Y', then D and D' are mutants iff Y and Y' are homeomorphic iff Y and Y' have isomorphic Heegaard Floer homology. The main input is a combinatorial result which characterizes the 2-isomorphism type of a graph in terms of the d-invariant of its lattice of integral flows.

The Jacobian variety of a non-singular curve is a basic tool in algebraic geometry, and a fundamental question is: "How to extend this construction to singular curves?" Starting with work of Igusa in the 1950's, a great deal of effort has gone into answering this question. Today we have a detailed understanding of how to assign a degenerate Jacobian to a singular curve, but our understanding of the geometry of the resulting object is less extensive. In my talk, I discuss work on the local geometry of the Caporaso-Pandharipande degenerate Jacobian. This work is joint with Sebastian Casalaina-Martin and Filippo Viviani.

In this talk we establish the existence and partial regularity of a (d-2)-dimensional edge-length minimizing polyhedron in $\R^d$. The minimizer is a generalized convex polytope of volume one which is the limit of a minimizing sequence of polytopes converging in the Hausdorff metric. We show that the (d-2)-dimensional edge-length $\zeta_{d-2}$ is lower-semicontinuous under this sequential convergence. Here the edge set of the limit generalized polytope is a closed subset of the boundary whose complement in the boundary consists of countably many relatively open planar regions.

By means of certain dispersive PDEs (such as the nonlinear Klein-Gordon equation) we will exhibit a new family of phenomena related to the ground state solitons. These solitons are exponentially unstable, and one can construct stable, unstable, and center(-stable) manifolds associated with these ground states in the sense of hyperbolic dynamics. In terms of these manifolds one can completely characterize the global dynamics of solutions whose energy exceeds that of the ground states by at most a small amount. In particular, we will establish a trichotomy in forward time giving either finite-time blow up, global forward existence and scattering to zero, or global existence and scattering to the ground states as all possibilities. It turns out that all nine sets consisting of all possible combinations of the forward/backward trichotomies arise. This extends the classical Payne-Sattinger picture (from 1975) which gives such a characterization at energies below that of the ground state; in the latter case the aforementioned (un)stable and center manifolds do not arise, since they require larger energy than that of the ground state. Our methods proceed by combining a perturbative analysis near the ground states with a global and variational analysis away from them. Most of this work is joint with Kenji Nakanishi from Kyoto University, Japan.

Consider singular submanifolds of Euclidean space admitting a generalized mean curvature. It is shown that this condition entails the existence of a second order structure. The proof uses approximation by Almgren's multiple-valued functions and a new differentiability result for Laplace's operator.

By a point of gradient catastrophe we mean a point where the leading order asymptotic behavior loses smoothness (for example, derivatives are not square integrable). In the case of the small dispersion (semiclassical) focusing NLS, this is a point where a slowly modulated high frequency plane wave wave suddenly burst into rapid amplitudial oscillation (spikes). Adjusting the nonlinear steepest descent (Deift-Zhou) method for Riemann-Hilbert problems, we give complete description of the leading order term near the point of gradient catastrophe in terms the {\em tritronqu\'ee} solution to the Painlev\'e I and rational breathers for the NLS. In fact, each spike corresponds to a pole of the {\em tritronqu\'ee} and has the universal shape shape of a scaled rational breather. Similar phenomenon was recently described for the asymptotic of orthogonal polynomials with complex varying weight $e^{-N (\hf z^2 + \qt t z^4)}$ near the critical value of the parameter $t$. In this case the spikes in the asymptotics for the recurrence coefficients turn out to be bounded for $t\in\R$ but unbounded for complex $t$.

4:00 pm Friday, April 8, 2011Thesis Defense: The (n)-solvable filtration of the link concordance group and Milnor's μ-invariants
by Carolyn Otto (Rice University) in HB 427

We establish several new results about the (n)-solvable filtration, {F mn }, of the string link concordance group Cm. We first establish a relationship between (n)-solvability of a link and its Milnor's μ-invariants. We study the effects of the Bing doubling operator on (n)-solvability. Using this results, we show that the "other half" of the filtration, namely F mn.5 / F mn+1 , is nontrivial and contains an infinite cyclic subgroup for links with sufficiently many components. We will also show that links modulo (1)-solvability is a nonabelian group. Lastly, we prove that the Grope filtration, G mn of Cm is not the same as the (n)-solvable filtration.

In this talk, we introduce the approximate converse theorem for globally unramified cuspidal representations of PGL(n, A), n>1. For a given set of Langlands parameters for some places of Q, we can compute epsilon>0 explicitly, which ensures that there eixsts a genuine unramified cuspidal representation within the boundary epsilon of the finite subset of the given parameters.

The celebrated theorems of Delorme (1977) and Guichardet (1972) establish the equivalence between property (T) and the vanishing of 1-cohomology, where the coefficients are taken in a unitary representation. In 2000 Shalom proved that the (a priori) weaker condition of the vanishing of reduced 1-cohomology is in fact equivalent to property (T) for the class of compactly generated groups. In 2005-2006 de Cornulier, Jolissaint, and Fernos independently showed that the vanishing of the restriction map on 1-cohomology is equivalent to relative property (T). One may ask if the relative version of Shalom's theorem is true. In a joint work with Valette we exhibit a large class of non-compact amenable group-pairs where the restriction map on reduced 1-cohomology always vanishes. Since amenable groups can not have relative property (T) with respect to non-compact subgroups, our result gives a strong negative answer to the above question.

Infinite Bernoulli convolution measures have been studied since the 1930's by Wintner, Erdos, Salem, Kahane, Garsia, and many other mathematicians. They first appeared in harmonic analysis and number theory, but more recently in other fields, in particular, dynamical systems and fractal geometry, as well as in applications. The talk will be a survey of the highlights of this topic, including some recent results and open problems.

4:00 pm Friday, April 15, 2011Colloquium: Developments on the multiple return time theorem
by Idris Assani (UNC) in HB 227

In 1991 we asked two questions on J. Bourgain return times theorem: The multiple term return time theorem and the break of duality. In this talk we will highlight the impact of these questions on more recent results in the theory of non conventional ergodic averages and generalizations of Carleson Hunt theorem on the a.e. convergence of Fourier Series.

We investigate the algebraic structure of knot Floer homology in the context of categorification. Ozsvath and Szabo gave the first completely algebraic description of knot Floer homology via a cube of resolutions construction. We use this construction to give a fully algebraic proof of invariance for knot Floer homology that avoids any mention of holomorphic disks or grid diagrams. We then reframe Ozsvath and Szabo's construction in the language of Soergel bimodules, which are the main ingredient in Khovanov's HOMFLY-PT homology, and explore the similarities between the two theories.

Consider the following basic problem about multiplication of polynomials in one variable. Fix a general 3-dimensional subspace V of the polynomials of degree a, and fix a second degree b. Given a subspace W of the polynomials of degree b, think of W as occupying the fraction dim(W)/(b+1) of the space of polynomials of degree b. For every such subspace W, does the product VW occupy at least as large a fraction of the polynomials of degree a+b as W does of the polynomials of degree b? That is, does multiplication by V always increase the fraction of the space occupied by W? Surprisingly, the answer to this question is connected to the golden ratio and its continued fraction expansion. We will further discuss how this question is connected with semistability and splitting properties of certain particularly nice vector bundles on P^2, known as Steiner bundles. These bundles can be viewed as natural generalizations of the tangent bundle. Finally, we will discuss how these bundles give rise to extremal effective divisors on the Hilbert scheme of points in P^2.

Spectral synthesis is a topic that was discussed intensively in the early seventies, but which was marked by numerous counterexamples. It had very few remaining problems after that time. I will discuss spectral synthesis for $H^1$, and prove a number of positive results including a characterization of compact sets of spectral synthesis for $H^1$.

In this talk we will give an approach to the following theorem: Let $\Gamma$ be an irreducible lattice in a connected semi-simple Lie group with finite center, no non-trivial compact factor and of rank bigger than one. Let $a:\Gamma \to Diff(T^N)$ be a real analytic action on the torus preserving an ergodic large measure (large means essentially that its support is non trivial in homotopy). $a$ induces a representation $a_0:\Gamma \to SL(N,Z)$. Assume further that $a_0$ has no zero weight and no rank one factor. Then $a$ and $a_0$ are conjugated by a real analytic map outside a finite $a_0$ invariant set. The theorem essentially says that nonlinear action $a$ is built from linear $a_0$ by blowing up finitely many point. This is joint work with A. Gorodnik, B. Kalinin and A. Katok.

Given an ergodic Jacobi matrix, there is a natural conformal map (DOS + i Lyapunov exponent) onto a slit domain. This tool was first used by Marchenko and Ostrowski in their study of periodic problems. In this talk, I'd like to give an overview of the situation for general ergodic (or just invariant) models. This is joint work (in progress) with Injo Hur.

Random symmetric matrices with entries that vanish outside a band around the diagonal, but otherwise have independent identically distributed matrix elements, were introduced in the physics literature as an effective model of a ”localization/delocalization” transition seen in disordered materials. In this talk, it will be shown that such matrices satisfy a localization condition which guarantees that eigenvectors have strong overlap with only $W^\mu$ standard basis vectors where W is the band width and $\mu$ is a positive exponent. Thus if $W^\mu << N$, with $N$ the size of the matrix, then a typical eigenvector is essentially supported on a vanishing fraction of standard basis vectors. Some open problems and conjectures will also be discussed.